Introduction to Spatial Computing CSE 555 Fundamental Spatial Concepts Some slides adapted from Worboys and Duckham (2004) GIS: A Computing Perspective, Second Edition, CRC Press

Logical Data Model for Spatial Data

Two Types of Models Field-based model: Contains a function from a spatial framework to an attribute domain Patterns of topographic altitudes, rainfall, etc. Object-based model: treats the space as populated by discrete, identifiable entities each with a geospatial reference Buildings, roads, etc.

Sample Raw Data Tuples recording annual weather conditions at different locations The field-based and object-based approaches are attempts to impose structure and pattern on such data.

Field-based approach Treats information as a collection of fields Each field defines the spatial variation of an attribute as a function from the set of locations to an attribute domain

Object-based Approach Clumps a relation as single or groups of tuples Certain groups of measurements of climatic variables can be grouped together into a finite set of types

Field Based Models

Spatial Framework Spatial framework: a partition of a region of space, forming a finite tessellation of spatial objects In the plane, the elements of a spatial framework are polygons Must be a finite structure for computational purposes Often the application domain will not be finite and sampling is necessary. Imprecision is introduced by the sampling process.

Spatial Fields Field Functions: f: Spatial Framework  Attribute Domain If the spatial framework is a Euclidean plane and the attribute domain is a subset of the set of real numbers; The Euclidean plane plays the role of the horizontal xy-plane The spatial field values give the z-coordinates, or “heights” above the plane

Layers in a GIS Can be considered as a the combination of the spatial framework and the field that assigns values for each location in the framework There may be many layers in a spatial database

Properties of Attribute Domain The attribute domain may contain values which are commonly classified into four levels of measurement Nominal attribute: simple labels; qualitative; cannot be ordered; and arithmetic operators are not permissible Ordinal attribute: ordered labels; qualitative; and cannot be subjected to arithmetic operators, apart from ordering Interval attributes: quantities on a scale without any fixed point; can be compared for size, with the magnitude of the difference being meaningful; the ratio of two interval attributes values is not meaningful Ratio attributes: quantities on a scale with respect to a fixed point; can support a wide range of arithmetical operations, including addition, subtraction, multiplication, and division

Continuous and differentiable fields Continuous field: small changes in location leads to small changes in the corresponding attribute value Differentiable field: rate of change (slope) is defined everywhere Spatial framework and attribute domain must be continuous for both these types of fields Every differentiable field must also be continuous, but not every continuous field is differentiable

One dimensional examples Fields may be plotted as a graph of attribute value against spatial framework Continuous and differentiable; the slope of the curve can be defined at every point

One dimensional examples The field is continuous (the graph is connected) but not everywhere differentiable. There is an ambiguity in the slope, with two choices at the articulation point between the two straight line segments. Continuous and not differentiable; the slope of the curve cannot be defined at one or more points

One dimensional examples The graph is not connected and so the field in not continuous and not differentiable. Not Continuous and not differentiable;

Two dimensional examples The slope is dependent on the particular location and on the bearing at that location

Operations on Fields A field operation takes as input one or more fields and returns a resultant field The system of possible operations on fields in a field-based model is referred to as map algebra Three main classes of operations Local Focal Zonal

Local Operations Local operation: acts upon one or more spatial fields to produce a new field The value of the new field at any location is dependent on the values of the input field function at that location.

Neighborhood function Given a spatial framework F, a neighborhood function n is a function that associates with each location x a set of locations that are “near” to x

Focal Operations Focal operation: the attribute value derived at a location x may depend on the attributes of the input spatial field functions at x and the attributes of these functions in the neighborhood n(x) of x

Zonal Operations Zonal operation: aggregates values of a field over a set of zones (arising in general from another field function) in the spatial framework For each location x: Find the Zone Zi in which x is contained Compute the values of the field function f applied to each point in Zi Derive a single value ζ(x) of the new field from the values computed in step 2

Question Consider a spatial framework defined by latitudes and longitude (1 degree by 1 degree cells) to represent (time-average) annual temperatures for years 1900 to present over the surface of Earth. Zones should be defined accordingly. Assume that each cell belongs to a unique zone. Classify following operations into local, focal, zonal, or a combination of these operations: Determine warmest temperature (or year) for each cell. Determine warmest cell in each country in year 2000. Identify country with highest average cell temperature in year 2000. For each cell, compute spatial-neighborhood average temperature in the year 2000. For each cell, compute heat-island-factor as the difference between its temperature and its spatial-neighborhood average temperature for the year 2000. Assume the results of previous step were available as an input for this step. Compute average annual temperature of surface of Earth for each year.

Spatial Auto-Correlation Spatial autocorrelation is a quantitative expression of Tobler’s first law of geography (1970) “Everything is related to everything else, but near things are more related than distant thing” Spatial autocorrelation measures the degree of clustering of values in a spatial field If like values tend to be located away from each other, then there is negative spatial autocorrelation If like values tend to cluster together, then the field exhibits high positive spatial autocorrelation If there is no apparent relationship between attribute value and location then there is zero spatial autocorrelation

Object Based Models

Entities Object-based models decompose an information space into objects or entities. An entity must be: Identifiable Relevant (be of interest) Describable (have characteristics) The frame of spatial reference is provided by the entities themselves.

House Object Has several attributes, such as registration date, address, owner and boundary, which are themselves objects

House Object The actual values of these attributes are literals If the house is registered to a new owner, we may change the registration attribute to a new date, however, the date November 5th, 1994” still exists as a date

Spatial objects Spatial objects are called “spatial” because they exist inside “space”, called the embedding space A set of primitive objects can be specified. Out of these all others in the application domain can be constructed, using an agreed set of operations Point-line-polygon primitives are common in existing systems.

GIS Analysis Query: For Italy’s capital city, Rome, calculate the total length of the River Tiber which lies within 2.5 km of the Colosseum. First we need to model the relevant parts as objects. Operation length will act on arc, and intersect will apply to form the piece of the arc in common with the disc

GIS Analysis A process of discretization must convert the objects to types that are computationally tractable A circle may be represented as a discrete polygonal area, Arcs by chains of line segments, Points may be embedded in some discrete space

Hierarchy of Spatial object types The most general spatial object type spatial is at the top of the hierarchy Spatial type is the disjoint union of types point and extent Class extent may be specialized by dimension into types 1- extent and 2- extent Two sub types of the one dimensional extents are described as arc and loop, specializing to simple arc and simple loop when there are no self-crossings

Hierarchy of Spatial object types The fundamental areal object is area A connected area is a region A region that is simply connected (no holes) is a cell

Topological Operators Object types with an assumed underlying topology are point, arc, loop and area Broadly three kinds Operations: (1) Topological: boundary, interior, closure and connected are defined in the usual manner is within provides a relationship between a point and a simple loop, returning true if the point is enclosed by the loop …… Many more are possible

Topological Operators Object types with an assumed underlying topology are point, arc, loop and area Broadly three kinds Operations: (1) Set Oriented: E.g., Intersection, union, subset of, etc. (2) Euclidian: E.g., distance, length, perimeter, centroid, etc.

Topological Spatial Operations for areas X meets Y if X and Y touch externally in a common portion of their boundaries X overlaps Y if X and Y impinge into each other’s interiors

Topological Spatial Operations for areas X is inside Y if X is a subset of Y and X, Y do not share a common portion of boundary X covers Y if Y is a subset of X and X, Y touch externally in a common portion of their boundaries

A “common” framework for Topological Operators Fundamental Relationships between X and Y boundary(X) ∩ boundary(Y) Interior(X) ∩ interior(Y) boundary(X) ∩ Interior(Y) Interior(X) ∩ boundary(Y) Interpretation 1 Meets Overlaps X inside Y … …. Fill this table and provide interpretations in the right column

A “common” framework for Topological Operators Fundamental Relationships between X and Y boundary(X) ∩ boundary(Y) Interior(X) ∩ interior(Y) boundary(X) ∩ Interior(Y) Interior(X) ∩ boundary(Y) Interpretation … …. These can be considered as a rough equivalent to relational operators used to model relational queries.

Nine intersection model of Topological Relationships Many toplogical Relationship between A and B can be specified using 9 intersection model Examples on next slide Nine intersections intersections between interior, boundary, exterior of A, B A and B are spatial objects in a two dimensional plane. Can be arranged as a 3 by 3 matrix Matrix element take a value of 0 (false) or 1 (true).

Nine intersection model of Topological Relationships